Geometric Mechanics, Part I

Dynamics And Symmetry

By: Darryl D Holm

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Paperback | 17 January 2008

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This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduate and beginning graduate students in mathematics, physics and engineering. It treats the dynamics of ray optics, resonant oscillators and the elastic spherical pendulum from a unified geometric viewpoint, by formulating their solutions using reduction by Lie-group symmetries. The only prerequisites are linear algebra, calculus and some familiarity with the Euler-Lagrange variational principles and canonical Poisson brackets in classical mechanics at the beginning undergraduate level.The ideas and concepts of geometric mechanics are explained in the context of explicit examples. Through these examples, the student develops skills in performing computational manipulations, starting from Fermat's principle, working through the theory of differential forms on manifolds and transferring these ideas to the applications of reduction by symmetry to reveal Lie-Poisson Hamiltonian formulations and momentum maps in physical applications.The many Exercises and Worked Answers aid the student to grasp the essential aspects of the subject. In addition, the modern language and application of differential forms is explained in the context of geometric mechanics, so that the importance of Lie derivatives and their flows is clear. All theorems are stated and proved explicitly.The book's many worked exercises make it ideal for both classroom use and self-study. In particular, a substantial appendix containing both introductory examples and enhanced coursework problems with worked answers is included to help the student develop proficiency in using the powerful methods of geometric mechanics.

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Both books are very readable; the author has an easy, informal style ]] These two books, written by one of the masters in geometric mechanics, provide an accessible way into the subject for newcomers; they also give a unique perspective for those who are not so new. -- Professor Peter Hydon "UK Nonlinear News Review"

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Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Science Science in General Maths for Scientists Applied Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Physics Classical Mathematics

Fluid Mechanics

Preface	p. xiii
Fermat's ray optics	p. 1
Fermat's principle	p. 1
Eikonal equation	p. 2
Huygens wave fronts	p. 5
The eikonal equation for mirages	p. 11
Hamiltonian formulation of ray optics	p. 12
Geometry, phase space and the ray path	p. 13
Legendre transformation	p. 15
Paraxial optics and classical mechanics	p. 18
Hamiltonian form of optical transmission	p. 19
Translation invariant media	p. 22
Axisymmetric, translation-invariant materials	p. 23
Hamiltonian optics equations, polar coordinates	p. 25
Geometric phase for Fermat's principle	p. 26
Skewness	p. 27
Lagrange invariant: Poisson bracket relations	p. 30
Axisymmetry	p. 33
Geometric picture in R[superscript 3]	p. 35
Flows of Hamiltonian vector fields	p. 37
Symplectic matrices	p. 41
Lie algebras	p. 46
Definitions	p. 46
Structure constants	p. 47
Commutator tables	p. 48
Poisson brackets among axisymmetric variables	p. 49
Non-canonical R[superscript 3] Poisson bracket for ray optics	p. 50
Equilibrium solutions	p. 52
Energy-Casimir stability	p. 52
Momentum maps	p. 55
The action of Sp(2, R) on T*R[superscript 2] [bsime] R[superscript 2] x R[superscript 2]	p. 55
Summary: Properties of momentum maps	p. 58
Lie-Poisson brackets	p. 61
The R[superscript 3]-bracket for ray optics is Lie-Poisson	p. 61
Lie-Poisson brackets with quadratic Casimirs	p. 62
Divergenceless vector fields	p. 66
Jacobi identity	p. 66
Geometric forms of Poisson brackets	p. 69
Nambu brackets	p. 71
Geometry of solution behaviour	p. 72
Restricting axisymmetric ray optics to level sets	p. 72
Geometric phase on level sets of S[superscript 2] = p[subscript o superscript 2]	p. 75
Singular ray optics in anisotropic media	p. 76
Ten geometrical features of ray optics	p. 80
Newton, Lagrange, Hamilton	p. 85
Newton	p. 85
Newton's Laws	p. 87
Dynamical quantities	p. 89
Newtonian form of free rigid rotation	p. 92
Lagrange	p. 95
Basic definitions for manifolds	p. 96
Euler-Lagrange equations on manifolds	p. 100
Geodesic motion on Reimannian manifolds	p. 104
Euler's equations for motion of a rigid body	p. 108
Hamilton	p. 111
Legendre transform	p. 112
Hamilton's canonical equations	p. 113
Phase space action principle	p. 114
Poisson brackets	p. 116
Canonical transformations	p. 117
Flows of Hamiltonian vector fields	p. 118
Properties of Hamiltonian vector fields	p. 120
Comparing approaches	p. 122
Comparing standard mechanics approaches	p. 122
Rigid-body motion	p. 124
Hamiltonian form of rigid-body motion	p. 124
Lie-Poisson Hamiltonian rigid-body dynamics	p. 125
Geometry of rigid-body level sets in R[superscript 3]	p. 127
Rotor and pendulum	p. 129
Reductions in the Maxwell-Bloch system	p. 134
Differential forms	p. 135
Symplectic manifolds	p. 135
Preliminaries for exterior calculus	p. 137
Manifolds and bundles	p. 137
Contraction	p. 138
Hamilton-Jacobi equation	p. 143
Lie derivatives	p. 144
Exterior calculus with differential forms	p. 144
Pull-back and push-forward notation	p. 146
Wedge product of differential forms	p. 147
Pull-back and push-forward of differential forms	p. 148
Summary of differential-form operations	p. 149
Contraction, or interior product	p. 150
Exterior derivative	p. 154
Exercises in exterior calculus operations	p. 155
Lie derivative	p. 158
Poincare's theorem	p. 158
Lie derivative exercises	p. 160
Ideal fluid dynamics	p. 162
Euler's fluid equations	p. 162
Lamb surfaces	p. 165
Helicity in incompressible fluids	p. 168
Silberstein-Ertel theorem for potential vorticity	p. 172
Hodge star operator on R[superscript 3]	p. 176
Poincare's Lemma	p. 179
Resonances and S[superscript 1] reduction	p. 185
Coupled oscillators on C[superscript 2]	p. 185
Oscillator variables in C[superscript 2]	p. 185
The 1:1 resonant action of S[superscript 1] on C[superscript 2]	p. 187
The S[superscript 1]-invariant Hermitian coherence matrix	p. 188
The Poincare sphere S[superscript 2] [set membership] S[superscript 3]	p. 189
1:1 resonance: Quotient map and orbit manifold	p. 191
The basic qubit: Quantum computing in the Bloch ball	p. 192
The action of SU (2) on C[superscript 2]	p. 194
Coherence dynamics for the 1:1 resonance	p. 197
Poisson brackets on the surface of a sphere	p. 198
Riemann projection of Poincare sphere	p. 199
Geometric and dynamic phases	p. 202
Geometric phase	p. 203
Dynamic phase	p. 203
Total phase	p. 204
Kummer shapes for n:m resonances	p. 205
Poincare map analog for n:m resonances	p. 209
The n:m resonant orbit manifold	p. 209
n:m Poisson bracket relations	p. 215
Nambu, or R[superscript 3]-bracket for n:m resonance	p. 216
Optical travelling-wave pulses	p. 217
Background	p. 217
Hamiltonian formulation	p. 219
Stokes vectors in polarisation optics	p. 220
Further reduction to the Poincare sphere	p. 223
Bifurcation analysis	p. 225
Nine regions in the ([lambda], [beta]) parameter plane	p. 227
Elastic spherical pendulum	p. 231
Introduction and problem formulation	p. 231
Problem statement, approach and results	p. 232
History of the problem	p. 232
Equations of motion	p. 233
Approaches of Newton, Lagrange and Hamilton	p. 233
Averaged Lagrangian technique	p. 243
A brief history of the three-wave equations	p. 246
A special case of the three-wave equations	p. 247
Reduction and reconstruction	p. 248
Phase portraits	p. 250
Geometry of the motion for fixed J	p. 252
Geometry of the motion for H = 0	p. 252
Three-wave surfaces	p. 253
Precession of the swing plane	p. 256
Maxwell-Bloch equations	p. 259
Self-induced transparency	p. 259
The Maxwell-Schrodinger Lagrangian	p. 260
Envelope approximation	p. 261
Averaged Lagrangian for envelope equations	p. 262
Complex Maxwell-Bloch equations	p. 263
Real Maxwell-Bloch equations	p. 264
Classifying Lie-Poisson structures	p. 265
Lie-Poisson structures	p. 266
Classes of Casimir functions	p. 268
Reductions to two dimensions	p. 271
Remarks on geometric phases	p. 275
Enhanced coursework	p. 279
Problem formulations	p. 279
Formulations of simple mechanical systems	p. 279
The bead sliding on a rotating hoop	p. 280
Equilibrium solutions	p. 284
Spherical pendulum	p. 285
Hamiltonian examples	p. 289
Criteria for canonical transformations	p. 289
Complex phase space: Oscillator variables	p. 292
2D resonant oscillators	p. 295
1:1 resonance	p. 296
1:-1 resonance	p. 300
The rigid rotor	p. 302
Examples of R[superscript 3] bracket dynamics	p. 304
Hamiltonian vector fields	p. 306
Lie derivatives and differential forms	p. 314
Elastic spherical pendulum & rigid body	p. 320
m[subscript 1]:m[subscript 2] Poincare sphere	p. 325
m[subscript 1]:m[subscript 2] resonant harmonic oscillators	p. 325
Multi-sheeted polar coordinates	p. 326
Resonant m[subscript 1]:m[subscript 2] torus	p. 327
Spherical pendulum: R[superscript 3] bracket	p. 328
Bibliography	p. 331
Index	p. 349
Table of Contents provided by Ingram. All Rights Reserved.

Geometric Mechanics, Part I

Dynamics And Symmetry

At a Glance

Paperback

Industry Reviews

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